Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f \in \mathscr { C } _ { b } ^ { 1 }$. We wish to show that $f$ admits a variance relative to $m$ and that $$\operatorname { Var } _ { m } ( f ) \leqslant \frac { C } { 2 } \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{2}$$ 10a. Show that $fm$ and $f ^ { 2 } m$ are integrable, and that it suffices to show (2) in the case where we additionally have $\int f ( x ) m ( x ) d x = 0$ and $\int f ( x ) ^ { 2 } m ( x ) d x = 1$. 10b. Under the hypotheses of the previous question, show (2). You may apply (1) to the family of functions $f _ { \varepsilon } = 1 + \varepsilon f$ for $\varepsilon > 0$.
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and
$$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$
Let $f \in \mathscr { C } _ { b } ^ { 1 }$. We wish to show that $f$ admits a variance relative to $m$ and that
$$\operatorname { Var } _ { m } ( f ) \leqslant \frac { C } { 2 } \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{2}$$
10a. Show that $fm$ and $f ^ { 2 } m$ are integrable, and that it suffices to show (2) in the case where we additionally have $\int f ( x ) m ( x ) d x = 0$ and $\int f ( x ) ^ { 2 } m ( x ) d x = 1$.
10b. Under the hypotheses of the previous question, show (2). You may apply (1) to the family of functions $f _ { \varepsilon } = 1 + \varepsilon f$ for $\varepsilon > 0$.